Question 1206240: So I really have no clue what to do for this question. The textbook asks for the coefficient of x. My thinking was that the coefficient should be 16, however, the answer is 96. The answer also calls for using Pascal's Triangle.
(2x+2)^4
Found 2 solutions by Theo, math_tutor2020:
Answer by Theo(13342)   (Show Source):
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
According to WolframAlpha, the x coefficient is 64 and not 96.
But the x^2 coefficient is 96. Perhaps your teacher made a mix up somehow.
(2x+2)^4 = 16x^4 + 64x^3 + 96x^2 + 64x + 16
https://www.wolframalpha.com/input?i=%282x%2B2%29%5E4
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(2x+2)^4 has the exponent 4, so we'll look at the row starting with "1,4,..." in Pascal's Triangle.
That full row of values is "1, 4, 6, 4, 1"
We'll multiply each of those items with powers of (2x) and (2)
The (2x) will have its exponent start at 4 and count down to 0; meanwhile the exponent for (2) will start at 0 and count up to 4.
Here's what it would look like
1*(2x)^4*(2)^0+4*(2x)^3*(2)^1+6*(2x)^2*(2)^2+4*(2x)^1*(2)^3+1*(2x)^0*(2)^4
Or perhaps it might be best to break things up line by line like this:
1*(2x)^4*(2)^0
4*(2x)^3*(2)^1
6*(2x)^2*(2)^2
4*(2x)^1*(2)^3
1*(2x)^0*(2)^4
The values in blue are the values mentioned in Pascal's Triangle.
The exponents for (2x) count down while the other (2) has the exponents counting up.
Notice for any monomial term the exponents add to 4.
Eg: 4*(2x)^3*(2)^1 has the exponents add to 3+1 = 4.
The x term has exponent 1 over the x, so it must have exponent 1 over the (2x) as well.
Focus on 4*(2x)^1*(2)^3 which would then simplify to 64x
Therefore, the answer is 64 assuming your teacher wants the x coefficient.
If s/he wanted the x^2 coefficient, then focus on 6*(2x)^2*(2)^2 which simplifies to 96x^2
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